Shubnikov–de Haas oscillations in the anomalous Hall conductivity of Chern insulators

The Haldane model on a honeycomb lattice is a paradigmatic example of a system featuring quantized Hall conductivity in the absence of an external magnetic field, that is, a quantum anomalous Hall effect. Recent theoretical work predicted that the anomalous Hall conductivity of massive Dirac fermions can display Shubnikov–de Haas (SdH) oscillations, which could be observed in topological insulators and honeycomb layers with strong spin-orbit coupling. Here, we investigate the electronic transport properties of Chern insulators subject to high magnetic fields by means of accurate spectral expansions of lattice Green's functions. We find that the anomalous component of the Hall conductivity displays visible SdH oscillations at low temperature. The effect is shown to result from the modulation of the next-nearest-neighbor flux accumulation due to the Haldane term, which removes the electron-hole symmetry from the Landau spectrum. To support our numerical findings, we derive a long-wavelength description beyond the linear (“Dirac cone”) approximation. Finally, we discuss the dependence of the energy spectra shift for reversed magnetic fields with the topological gap and the lattice bandwidth.

Figure 1

(a) Hopping phases of the Peierls substitution with a Landau gauge $\mathbf{A}=-By\widehat{x}$ in the unit cell $(n,m)$. Phases for the NN hopping are in green. Phases for the NNN hopping in sublattices $A$ and $B$ are in red and blue, respectively. Here, ${\varphi }^{\prime }=3\sqrt{3}{a}^{2}eB/4\hslash .$ (b) Fluxes enclosed by the different paths involving NN and NNN hoppings for positive $B$.

Figure 2

Density of states calculated as a function of energy for a system described by the Haldane model with $2\times 800\times 400$ sites, $t_2=\frac{0.5t}{6\sqrt{3}},B=131\text{T}$, and $\gamma =0.1t$. The spectral expansion [Eq. (2)] employed 4000 polynomials and 70 random vectors. A Jackson kernel was used to damp Gibbs oscillations [24]. (a) Plot of the full density of states for positive field configurations (blue dashed line) and negative field configurations (red solid line). (b) Closeup of the DOS around the $n=1$ and $n=2$ LLs.

Figure 3

Hall conductivity of the Haldane model calculated as a function of energy. Simulation parameters as in Fig. 2. Panels (a) and (b) depict the Hall conductivities calculated for $\vec{B}=\mp B\widehat{z}$, respectively, and panel (c) shows the calculated anomalous part of the Hall conductivity.

Figure 4

Energy spectra difference $\mathrm{\Delta }{E}_n=E_n^{\left(1\right)}-E_n^{\left(2\right)}$ calculated for positive (red circles) and negative (blue squares) magnetic field. Inset: energy spectra shift according to continuum model (red line) and tight-binding calculations (solid circles). Simulation parameters: $t_2=\frac{0.2t}{6\sqrt{3}},D=2\times {10}^6$ atoms, $\left|B\right|=157\text{T},N=5000$, and $R=60$.

Figure 5

(a) Anomalous part of the Hall conductivity calculated for the Haldane model in the continuum limit, including second order corrections. Results are obtained for $t_2\approx 0.03$ eV, $\left|B\right|=10$ T, $\mathrm{\Gamma }=0.3$ meV, and different temperatures: $T=35$ K (blue solid line), $T=70$ K (red dashed line), and $T=140$ K (black solid line). Panel (b) highlights the SdH oscillations in the anomalous part of the Hall conductivity.